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7.0   Special Cubes, Prime Numbers

7.7   Associated Magic Cubes (5 x 5 x 5)

7.7.1 Introduction

Associated Magic Cubes of order 5 can be considered as being composed out of an Associated Border and a (non magic) Associated Inner Cube.

7.7.2 (Semi-) Magic Surface Planes

Associated Magic Cubes can be constructed based on Complementary Anti Symmetric Magic Squares of order 5, as discussed in Section 14.3.10.

The construction method, based on this principle, can be summarized as follows:

• Construct Anti Symmetric Magic Squares of order 5, which can be considered as possible top squares for the Associated Border (ref. Attachment 7.7.2);
• Determine, based on the selected top - and resulting bottom square, the back - and front square;
• Determine, based on the top -, bottom -, back - and front squares, the left - and resulting right square.

The relation between opposite surface squares (symmetry) can be represented as follows:

 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c20 c21 c22 c23 c24 c25
 Pr5 - c25 Pr5 - c24 Pr5 - c23 Pr5 - c22 Pr5 - c21 Pr5 - c20 Pr5 - c19 Pr5 - c18 Pr5 - c17 Pr5 - c16 Pr5 - c15 Pr5 - c14 Pr5 - c13 Pr5 - c12 Pr5 - c11 Pr5 - c10 Pr5 - c9 Pr5 - c8 Pr5 - c7 Pr5 - c6 Pr5 - c5 Pr5 - c4 Pr5 - c3 Pr5 - c2 Pr5 - c1

with Pr5 = 2 * s5 / 5 the pair sum for the corresponding Magic Sum s5.

With c(i) the cube variables and the substitution:

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25)
=
 c(1) c(2) c(3) c(4) c(5) c(26) c(27) c(28) c(29) c(30) c(51) c(52) c(53) c(54) c(55) c(76) c(77) c(78) c(79) c(80) c(101) c(102) c(103) c(104) c(105)

the defining equations of the Back Square (Semi Magic) can be written as:

```a( 6) = s5 - a(11) - a(16) - a( 1) - a(21)
a( 7) = s5 - a(12) - a(17) - a( 2) - a(22)
a( 8) = s5 - a(13) - a(18) - a( 3) - a(23)
a( 9) = s5 - a(14) - a(19) - a( 4) - a(24)
a(10) = s5 - a(15) - a(20) - a( 5) - a(25)
a(11) = s5 - a(12) - a(13) - a(14) - a(15)
a(16) = s5 - a(17) - a(18) - a(19) - a(20)
```

with a(i) independent for i = 12 ... 15 and i = 17 ... 20;
and  a(i) defined     for i =  1 ...  5 and i = 21 ... 25.

Based on a comparable substitution:

 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25)
=
 c(1) c(6) c(11) c(16) c(21) c(26) c(31) c(36) c(41) c(46) c(51) c(56) c(61) c(66) c(71) c(76) c(81) c(86) c(91) c(96) c(101) c(106) c(111) c(116) c(121)

the defining equations of the Left Square (Semi Magic) can be written as:

```a( 7) = s5 - a(12) - a(17) - a( 2) - a(22)
a( 8) = s5 - a(13) - a(18) - a( 3) - a(23)
a( 9) = s5 - a(14) - a(19) - a( 4) - a(24)
a(12) = s5 - a(13) - a(14) - a(11) - a(15)
a(17) = s5 - a(18) - a(19) - a(16) - a(20)
```

with a(i) independent for i = 13, 14, 18 and 19;
and  a(i) defined     for i = 1 ... 6, 10, 11, 15, 16 and 20 ... 25.

The defining equations of the Associated Inner Cube can be written as:

```c(92) = s5 - c(91) - c(93) - c(94) - c(95)
c(84) = s5 - c(79) - c(89) - c(94) - c(99)
c(87) = s5 - c(86) - c(88) - c(89) - c(90)
c(83) = s5 - c(78) - c(88) - c(93) - c(98)
c(82) = s5 - c(81) - c(83) - c(84) - c(85)
c(69) = s5 - c(19) - c(44) - c(94) - c(119)
c(68) = s5 - c(18) - c(43) - c(93) - c(118)
c(67) = s5 - c(17) - c(42) - c(92) - c(117)
c(64) = s5 - c(14) - c(39) - c(89) - c(114)
```
 c(32) = Pr5 - c(94) c(33) = Pr5 - c(93) c(34) = Pr5 - c(92) c(37) = Pr5 - c(89) c(42) = Pr5 - c(84) c(38) = Pr5 - c(88) c(39) = Pr5 - c(87) c(43) = Pr5 - c(83) c(44) = Pr5 - c(82) c(57) = Pr5 - c(69) c(58) = Pr5 - c(68) c(59) = Pr5 - c(67) c(62) = Pr5 - c(64)

with c(i) independent for i = 88, 89, 93 and 94;
and  c(i) defined     for i = 14, 17 ... 19, 78, 79, 81, 85, 86, 90, 93, 95, 98. 99, 114 and 117 ... 119

Based on the equations listed above, a guessing routine can be written to generate Prime Number Associated Magic Cubes of order 5 within a reasonable time (PrimeCubes5d).

Attachment 7.7.3, shows the first occurring 5th order Prime Number Associated Magic Cube for miscellaneous Magic Sums

7.7.4 Summary

The obtained results regarding the miscellaneous Prime Number Associated Magic Cubes as deducted and discussed in previous sections are summarized in following table:

 Order Main Characteristics Subroutine Results 5 Associated, (Semi-) Magic Surface Planes - - - -
 Comparable routines as listed above, can be used to generate Prime Number Perfect Concentric Magic Cubes, which will be described in following sections.